Electric-field-gradient analysis of high-$T_c$ superconductors

The qualitative and quantitative features of the electronic charge distribution in cuprate superconductors are reexamined on the basis of a comprehensive set of nuclear quadrupole resonance and NMR measurements. A systematic analysis of measured electric-field gradients is performed within the tight-binding approach, commonly used for electronic models in cuprates. Both in-plane and out-of-plane sites and orbitals are considered. Special attention is given to the generic $\mathrm{pd}$ model involving only the $\sigma$ orbitals in the ${\mathrm{CuO}}_2$ planes. This model is checked against the experimental data. The physical origin of several material-dependent features is clarified. It is shown that in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ the ${3d}_{{3z}^2-r^2}$ orbital of the in-plane copper and the ${2p}_z$ apex oxygen orbital are occupied by a substantially smaller fraction of holes than the in-plane $\sigma$ orbitals, at variance with proposals lending importance to copper-apex oxygen hybridization. At the in-plane oxygen site a sizable ${2p}_z$ admixture is found at the Fermi level of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$. The in-plane charge distribution of ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_4$ is found to be qualitatively similar to that of the high-temperature tetragonal phase of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$. It appears that the additional holes are shared among copper and oxygen sites in similar proportions. This is shown to be compatible with the large $U_d$ Emery model provided that the difference between the $p$ and $d$ atomic energies is comparable to the first neighbor overlap energy.